In mathematics, an amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements. The original definition, in terms of a finitely additive invariant measure (or mean) on subsets of G, was introduced by John von Neumann in 1929 under the German name "messbar" ("measurable" in English) in response to the Banach–Tarski paradox. In 1949 Mahlon M. Day introduced the English translation "amenable", apparently as a pun.
The amenability property has a large number of equivalent formulations. In the field of analysis, the definition is in terms of linear functionals. An intuitive way to understand this version is that the support of the regular representation is the whole space of irreducible representations.
If a group has a Følner sequence then it is automatically amenable.
Definition for locally compact groups
Let G be a locally compact Hausdorff group. Then it is well known that it possesses a unique, up-to-scale left- (or right-) rotation invariant ring (borel regular in the case of G second countable) measure (left and right probability measure in the case of G compact), the Haar measure. Consider the banach space L∞(G) of essentially-bounded measurable functions within this measure space (which is clearly independent of the scale of the Haar measure).
Definition 1. A linear functional Λ in Hom(L∞(G), R) is said to be a mean if Λ has norm 1 and is non-negative, i.e. f ≥ 0 a.e. implies Λ(f) ≥ 0.
Definition 2. A mean Λ in Hom(L∞(G), R) is said to be left-invariant (resp. right-invariant) if Λ(g·f) = Λ(f) for all g in G, and f in L∞(G) with respect to the left (resp. right) shift action of g·f(x) = f(xg−1)(resp. f(x)·g = f(g−1x) ).
Definition 3. A locally compact Hausdorff group is called amenable if it admits a left- (or right-)invariant mean.
Equivalent conditions for amenability
- Existence of a left (or right) invariant mean on L∞(G). The original definition, which depends on the axiom of choice.
- Existence of left-invariant states. There is a left-invariant state on any separable left-invariant unital C* subalgebra of the bounded continuous functions on G.
- Fixed-point property. Any action of the group by continuous affine transformations on a compact convex subset of a (separable) locally convex topological vector space has a fixed point. For locally compact abelian groups, this property is satisfied as a result of the Markov–Kakutani fixed-point theorem.
- Irreducible dual. All irreducible representations are weakly contained in the left regular representation λ on L2(G).
- Trivial representation. The trivial representation of G is weakly contained in the left regular representation.
- Godement condition. Every bounded positive-definite measure μ on G satisfies μ(1) ≥ 0. Valette (1998) improved this criterion by showing that it is sufficient to ask that, for every continuous positive-definite compactly supported function f on G, the function Δ–½f has non-negative integral with respect to Haar measure, where Δ denotes the modular function.
- Day's asymptotic invariance condition. There is a sequence of integrable non-negative functions φn with integral 1 on G such that λ(g)φn − φn tends to 0 in the weak topology on L1(G).
- Reiter's condition. For every finite (or compact) subset F of G there is an integrable non-negative function φ with integral 1 such that λ(g)φ − φ is arbitrarily small in L1(G) for g in F.
- Dixmier's condition. For every finite (or compact) subset F of G there is unit vector f in L2(G) such that λ(g)f − f is arbitrarily small in L2(G) for g in F.
- Glicksberg−Reiter condition. For any f in L1(G), the distance between 0 and the closed convex hull in L1(G) of the left translates λ(g)f equals |∫f|.
- Følner condition. For every finite (or compact) subset F of G there is a measurable subset U of G with finite positive Haar measure such that m(U Δ gU)/m(U) is arbitrarily small for g in F.
- Leptin's condition. For every finite (or compact) subset F of G there is a measurable subset U of G with finite positive Haar measure such that m(FU Δ U)/m(U) is arbitrarily small.
- Kesten's condition. Left convolution on L2(G) by a probability measure on G gives an operator of operator norm 1.
- Johnson's cohomological condition. The Banach algebra A = L1(G) is amenable as a Banach algebra, i.e. any bounded derivation of A into the dual of a Banach A-bimodule is inner.
Case of discrete groups
Definition. A discrete group G is amenable if there is a finitely additive measure (also called a mean) —a function that assigns to each subset of G a number from 0 to 1—such that
- The measure is a probability measure: the measure of the whole group G is 1.
- The measure is finitely additive: given finitely many disjoint subsets of G, the measure of the union of the sets is the sum of the measures.
- The measure is left-invariant: given a subset A and an element g of G, the measure of A equals the measure of gA. (gA denotes the set of elements ga for each element a in A. That is, each element of A is translated on the left by g.)
This definition can be summarized thus: G is amenable if it has a finitely-additive left-invariant probability measure. Given a subset A of G, the measure can be thought of as answering the question: what is the probability that a random element of G is in A?
It is a fact that this definition is equivalent to the definition in terms of L∞(G).
Having a measure μ on G allows us to define integration of bounded functions on G. Given a bounded function f : G → R, the integral
is defined as in Lebesgue integration. (Note that some of the properties of the Lebesgue integral fail here, since our measure is only finitely additive.)
If a group has a left-invariant measure, it automatically has a bi-invariant one. Given a left-invariant measure μ, the function μ−(A) = μ(A−1) is a right-invariant measure. Combining these two gives a bi-invariant measure:
The equivalent conditions for amenability also become simpler in the case of a countable discrete group Γ. For such a group the following conditions are equivalent:
- Γ is amenable.
- If Γ acts by isometries on a (separable) Banach space E, leaving a weakly closed convex subset C of the closed unit ball of E* invariant, then Γ has a fixed point in C.
- There is a left invariant norm-continuous functional μ on ℓ∞(Γ) with μ(1) = 1 (this requires the axiom of choice).
- There is a left invariant state μ on any left invariant separable unital C* subalgebra of ℓ∞(Γ).
- There is a set of probability measures μn on Γ such that ||g · μn − μn||1 tends to 0 for each g in Γ (M.M. Day).
- There are unit vectors xn in ℓ2(Γ) such that ||g · xn − xn||2 tends to 0 for each g in Γ (J. Dixmier).
- There are finite subsets Sn of Γ such that |g · Sn Δ Sn| / |Sn| tends to 0 for each g in Γ (Følner).
- If μ is a symmetric probability measure on Γ with support generating Γ, then convolution by μ defines an operator of norm 1 on ℓ2(Γ) (Kesten).
- If Γ acts by isometries on a (separable) Banach space E and f in ℓ∞(Γ, E*) is a bounded 1-cocycle, i.e. f(gh) = f(g) + g·f(h), then f is a 1-coboundary, i.e. f(g) = g·φ − φ for some φ in E* (B.E. Johnson).
- The von Neumann group algebra of Γ is hyperfinite (A. Connes).
Note that A. Connes also proved that the von Neumann group algebra of any connected locally compact group is hyperfinite, so the last condition no longer applies in the case of locally compact groups.
Amenability is related to the spectral problem of Laplacians. For instance, the fundamental group of a closed Riemannian manifold is amenable if and only if the bottom of the spectrum of the Laplacian is 0 (R. Brooks, T. Sunada).
- Every (closed) subgroup of an amenable group is amenable.
- Every quotient of an amenable group is amenable.
- A group extension of an amenable group by an amenable group is again amenable. In particular, finite direct product of amenable groups are amenable, although infinite products need not be.
- Direct limits of amenable groups are amenable. In particular, if a group can be written as a directed union of amenable subgroups, then it is amenable.
- Amenable groups are unitarizable; the converse is an open problem.
- Countable discrete amenable groups obey the Ornstein isomorphism theorem.
- Finite groups are amenable. Use the counting measure with the discrete definition. More generally, compact groups are amenable. The Haar measure is an invariant mean (unique taking total measure 1).
- The group of integers is amenable (a sequence of intervals of length tending to infinity is a Følner sequence). The existence of a shift-invariant, finitely additive probability measure on the group Z also follows easily from the Hahn–Banach theorem this way. Let S be the shift operator on the sequence space ℓ∞(Z), which is defined by (Sx)i = xi+1 for all x ∈ ℓ∞(Z), and let u ∈ ℓ∞(Z) be the constant sequence ui = 1 for all i ∈ Z. Any element y ∈ Y:=Ran(S − I) has a distance larger than or equal to 1 from u (otherwise yi = xi+1 - xi would be positive and bounded away from zero, whence xi could not be bounded). This implies that there is a well-defined norm-one linear form on the subspace Ru + Y taking tu + y to t. By the Hahn–Banach theorem the latter admits a norm-one linear extension on ℓ∞(Z), which is by construction a shift-invariant finitely additive probability measure on Z.
- By the direct limit property above, a group is amenable if all its finitely generated subgroups are. That is, locally amenable groups are amenable.
- By the fundamental theorem of finitely generated abelian groups, it follows that abelian groups are amenable.
- It follows from the extension property above that a group is amenable if it has a finite index amenable subgroup. That is, virtually amenable groups are amenable.
- Furthermore, it follows that all solvable groups are amenable.
- Finitely generated groups of subexponential growth are amenable. A suitable subsequence of balls will provide a Følner sequence.
If a countable discrete group contains a (non-abelian) free subgroup on two generators, then it is not amenable. The converse to this statement is the so-called von Neumann conjecture, which was disproved by Olshanskii in 1980 using his Tarski monsters. Adyan subsequently showed that free Burnside groups are non-amenable: since they are periodic, they cannot contain the free group on two generators. These groups are finitely generated, but not finitely presented. However, in 2002 Sapir and Olshanskii found finitely presented counterexamples: non-amenable finitely presented groups that have a periodic normal subgroup with quotient the integers.
For finitely generated linear groups, however, the von Neumann conjecture is true by the Tits alternative: every subgroup of GL(n,k) with k a field either has a normal solvable subgroup of finite index (and therefore is amenable) or contains the free group on two generators. Although Tits' proof used algebraic geometry, Guivarc'h later found an analytic proof based on V. Oseledets' multiplicative ergodic theorem. Analogues of the Tits alternative have been proved for many other classes of groups, such as fundamental groups of 2-dimensional simplicial complexes of non-positive curvature.
- Day's first published use of the word is in his abstract for an AMS summer meeting in 1949, Means on semigroups and groups, Bull. A.M.S. 55 (1949) 1054–1055. Many text books on amenabilty, such as Volker Runde's, suggest that Day chose the word as a pun.
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